On sufficiency and duality in multiobjective programming problem under generalized <Emphasis Type="Italic">α</Emphasis>-type I univexity

In this paper, we are concerned with the multiobjective programming problem with inequality constraints. We introduce new classes of generalized α-univex type I vector valued functions. A number of Kuhn–Tucker type sufficient optimality conditions are obtained for a feasible solution to be an efficient solution. The Mond–Weir type duality results are also presented.     

A formula for deficiency (L-andR-tests)

In this paper an analogue of the formulas [D. M. Chibisov,Teor. Veroyatn. Primen.,30, 269–288 (1985);Izv. Akad. Nauk UzSSR,6, 23–30 (1982)] for the difference between the power of a given asymptotically efficient test and that of the most powerful test is justified for one-sample L-and R-tests, i.e., tests based on linear combinations of order statistics and linear rank statistics. This formula directly yields the Hodges-Lehmann deficiency of corresponding tests. A general theorem is stated which is applied to L-and R-tests. The explicit expressions given by this formula for L- and R-tests are also presented. The expression related to R-tests agrees with the one obtained in [W. Albers, P. J. Bickel, and W. R. Van Zwet,Ann. Statist.,4, 108–156 (1976);6, 1170–1171 (1978)]. We present here a nontechnical (heuristic) proof of these results. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-1446). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

Fractional Sobolev,Moser–Trudinger,Morrey–Sobolev inequalities under Lorentz norms

We consider the Sobolev type inequalities under Lorentz norms on bounded open domains for fractional derivatives (−∆) ^s/2 in the following three cases: n > ps, n = ps, and n < ps, whence establishing the weak type Sobolev inequalities, Moser–Trudinger and Morrey–Sobolev inequalities for fractional derivatives in Lorentz norms. Applying these inequalities, we obtain the trace forms of six related functional inequalities. Bibliography: 44 titles.     

New Identities for Hall-Littlewood Polynomials and Applications

Starting from Macdonald's summation formula of Hall-Littlewood polynomials over bounded partitions and its even partition analogue, Stembridge (Trans. Amer. Math. Soc., 319(2), (1990) 469–498) derived sixteen multiple q-identities of Rogers–Ramanujan type. Inspired by our recent results on Schur functions (Adv. Appl. Math., 27, (2001) 493–509) and based on computer experiments we obtain two further such summation formulae of Hall-Littlewood polynomials over bounded partitions and derive six new multiple q-identities of Rogers–Ramanujan type. 2000 Mathematics Subject Classification: Primary–05A19; Secondary–05A17, 05A30     

Half-space type theorems in warped product spaces with one-dimensional factor

This work states some half-space type theorems in a warped product space of the form I ×_ρ M, where is an open interval and M is either a compact n-manifold, or a complete simply connected surface with constant curvature c ≤ 0. Such theorems generalize the classical half-space theorem for minimal surfaces in R ³, obtained by Hoffmann and Meeks (Invent Math 101:373–377, 1990), and recent results for surfaces contained in a slab of R ×_ρ M, obtained by Dajczer and Alías (Comment Math Helvetici 81:653–663, 2006).      

Enhanced interface repulsion from quenched hard–wall randomness

 We consider the harmonic crystal, or massless free field, , , that is the centered Gaussian field with covariance given by the Green function of the simple random walk on ℤ ^d . Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition to be larger than , is an IID field (which is also independent of ϕ), for every x in a large region , with N a positive integer and D a bounded subset of ℝ ^d . We are mostly motivated by results for given typical realizations of σ (quenched set–up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, living in a (d+1)–dimensional space, constrained not to go below an inhomogeneous substrate that acts as a hard wall. We consider various types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of σ ₀ is heavier than Gaussian, while essentially no effect is observed if the tail is sub–Gaussian. In the critical case, that is the one of approximately Gaussian tail, the interplay of the two sources of randomness, ϕ and σ, leads to an enhanced repulsion effect of additive type. This generalizes work done in the case of a flat wall and also in our case the crucial estimates are optimal Large Deviation type asymptotics as of the probability that ϕ lies above σ in D _N . Received: 6 February 2002 / Revised version: 23 May 2002 / Published online: 30 September 2002 Mathematics Subject Classification (2000): 82B24, 60K35, 60G15 Keywords or phrases: Harmonic Crystal – Rough Substrate – Quenched and Annealed Models – Entropic Repulsion – Gaussian fields – Extrema of Random Fields – Large Deviations – Random Walks     

On the unsteady Poiseuille flow in a pipe

We show that in an unsteady Poiseuille flow of a Navier–Stokes fluid in an infinite straight pipe of constant cross-section, σ, the flow rate, F(t), and the axial pressure drop, q(t), are related, at each time t, by a linear Volterra integral equation of the second type, where the kernel depends only upon t and σ. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given F(t) is equivalent to the resolution of the classical initial-boundary value problem for the heat equation. G. P. Galdi: Partially supported by the NSF grant DMS–0404834. K. Pileckas: Supported by EC FP6 MCToK program SPADE2, MTKD–CT–2004–014508 A. L. Silvestre: Supported by FCT-Project POCI/MAT/61792/2004     

The general gould type integral with respect to a multisubmeasure

In two earlier papers [GAVRILUŢ, A.: A Gould type integral with respect to a multisubmeasure, Math. Slovaca 58 (2008), 1–20] and [Gavriluţ, A.: On some properties of the Gould type integral with respect to a multisubmeasure, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 52 (2006), 177–194], we defined and studied a Gould type integral for a real valued, bounded function with respect to a multisubmeasure having finite variation. In this paper, we introduce and study the properties of a Gould type integral in the general setting: the function may be unbounded and the variation of the multisubmeasure may be infinite.     

Helmholtz potentiality conditions for systems of difference-differential equations

For operators corresponding to systems ofμth-order difference-differential equations of neutral type, necessary and sufficient conditions for potentiality with respect to the classical bilinear form are obtained. These results are applied to first-order quasilinear systems. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 437–442, September, 1998. The author wishes to express his thanks to V. M. Savchin for fruitful discussions and useful remarks.     

Limit theorems for Toeplitz quadratic functionals of continuous-time stationary processes

Let X(t), , be a centered real-valued stationary Gaussian process with spectral density f(λ). The paper considers a question concerning asymptotic distribution of Toeplitz type quadratic functional Q _T of the process X(t), generated by an integrable even function g(λ). Sufficient conditions in terms of f(λ) and g(λ) ensuring central limit theorems for standard normalized quadratic functionals Q _T are obtained, extending the results of Fox and Taqqu (Prob. Theory Relat. Fields 74: 213–240, 1987), Avram (Prob. Theory Relat. Fields 79:37–45, 1988), Giraitis and Surgailis (Prob. Theory Relat. Fields 86: 87–104, 1990), Ginovian and Sahakian (Theory Prob. Appl. 49:612–628, 2004) for discrete time processes.      

Comparison of ‘Tail Method’ Exact Confidence Limits for the Difference of Binomial Probabilities

Suppose that Y ₁ and Y ₂ are independent and have Binomial(n ₁,p ₁) and Binomial (n ₂,p ₂) distributions respectively. Also suppose that θ=p ₁−p ₂ is the parameter of interest. We consider the problem of finding an exact confidence limit (either upper or lower) for θ. The solution to this problem is very important for statistical practice in the health and life sciences. The ‘tail method’ provides a solution to this problem. This method finds the exact confidence limit by exact inversion of a hypothesis test based on a specified test statistic. Buehler (J. Am. Stat. Assoc. 52, 482–493, 1957) described, for the first time, a finite-sample optimality property of this confidence limit. Consequently, this confidence limit is sometimes called a Buehler confidence limit. An early tail method confidence limit for θ was described by Santner and Snell (J. Am. Stat. Assoc. 75, 386–394, 1980) who used the maximum likelihood estimator of θ as the test statistic. This confidence limit is known to be very inefficient (see e.g. Cytel Software, StatXact, version 6, vol. 2, 2004). The efficiency of the confidence limit resulting from the tail method depends greatly on the test statistic on which it is based. We use the results of Kabaila (Stat. Probab. Lett. 52, 145–154, 2001) and Kabaila and Lloyd (Aust. New Zealand J. Stat. 46, 463–469, 2004, J. Stat. Plan. Inference 136, 3145–3155, 2006) to provide a detailed explanation for the dependence of this efficiency on the test statistic. We consider test statistics that are estimators, Z-statistics and approximate upper confidence limits. This explanation is used to find the situations in which the tail method exact confidence limits based on test statistics that are estimators or Z-statistics are least efficient.     

On the errors of misclassification based on dichotomous and normal variables

Summary The distribution of the errors of misclassification in procedures based on dichotomous and normal variables is derived. The expressions forE(e ₁₂) andE(e ₂₁) are also obtained. The results in the paper extend those of Chang and Afifi (1974,J. Amer. Statist. Ass.,69, 336–339), using the earlier papers due to John (1961,Ann. Math. Statist.,32, 1125–1144), Subrahmaniam and Chinganda (1978,J. Statist. Plann. Inf.,2, 79–91).     

Likelihood ratio tests for comparingk populations —The two-parameter nonregular models

Summary The null and nonnull distributions of the likelihood ratio statistics for testing the homogeneity ofk given populations, each associated with a nonregular density depending on two truncation parameters, are investigated. This generalizes to the two-parameter case the work of Hogg (1956,Ann. Math. Statist.,27, 529–532), Barr (1966,J. Amer. Statist. Assoc.,61, 856–864) and Khatri and Jaiswal (1969,Aust. J. Statist.,11, 79–84; 1969, 1971,Ann. Inst. Statist. Math.,21, 127–136;23, 199–210).     

Some New Inverse-type Hilbert-Pachpatte Integral Inequalities

In this paper,some new generalizations of inverse type Hilbert-Pachpatte integral inequalities are proved.The results of this paper reduce to those of Pachpatte(1998,J.Math.Anal.Appl.226,166-179)and Zhan and Debnath(2001,J.Math.Anal.Appl.262,411-418).     

Localization transition of <Emphasis Type="Italic">d</Emphasis>-friendly walkers

 Friendly walkers is a stochastic model obtained from independent one-dimensional simple random walks {S ^k _j } _j≥0 , k=1,2,…,d by introducing ``non-crossing condition': and ``reward for collisions' characterized by parameters . Here, the reward for collisions is described as follows. If, at a given time n, a site in ℤ is occupied by exactly m≥2 walkers, then the site increases the probabilistic weight for the walkers by multiplicative factor exp (β _m )≥1. We study the localization transition of this model in terms of the positivity of the free energy and describe the location and the shape of the critical surface in the (d−1)-dimensional space for the parameters . Received: 13 June 2002 / Revised version: 24 August 2002 Published online: 28 March 2003 Mathematics Subject Classification (2000): 82B41, 82B26, 82D60, 60G50 Key words or phrases: Random walks – Random surfaces – Lattice animals – Phase transitions – Polymers – Random walks     

On Relative Difference Sets in Dihedral Groups

In this paper, we study extensions of trivial difference sets in dihedral groups. Such relative difference sets have parameters of the form (uλ,u,uλ, λ) or (uλ+2,u, uλ+1, λ) and are called semiregular or affine type, respectively. We show that there exists no nontrivial relative difference set of affine type in any dihedral group. We also show a connection between semiregular relative difference sets in dihedral groups and Menon–Hadamard difference sets. In the last section of the paper, we consider (m, u, k, λ) difference sets of general type in a dihedral group relative to a non-normal subgroup. In particular, we show that if a dihedral group contains such a difference set, then m is neither a prime power nor product of two distinct primes.     

Asymptotic behavior of functionals of empirical distribution functions for the two-sample problem

Summary Mises functional is extended for the two-sample problem. It is shown that the extended Mises functional also has the asymptotic property given by von Mises (1947,Ann. Math. Statist.,18, 309–348) and by Filippova (1962,Theory Prob. Appl.,7, 24–57) in the one-sample case. Asymptotic behavior ofU-statistic in the two-sample case, the statistic of Cramér-von Mises type for testing homogeneity and so forth are investigated as important examples of the theory. The Institute of Statistical Mathematics     

Leading coefficients of Kazhdan–Lusztig polynomials for Deodhar elements

We show that the leading coefficient of the Kazhdan–Lusztig polynomial P _x,w (q) known as μ(x,w) is always either 0 or 1 when w is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance in Billey and Warrington (J. Algebraic Combin. 13(2):111–136, [2001]) and Billey and Jones (Ann. Comb. [2008], to appear). In type A, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar’s algorithm (Deodhar in Geom. Dedicata 63(1):95–119, [1990]), we provide some combinatorial criteria to determine when μ(x,w)=1 for such permutations w. The author received support from NSF grants DMS-9983797 and DMS-0636297.     

The Riemann problem on a finite-sheeted Riemann surface of infinite genus

LetR be the Riemann surface of the functionu(z) specified by the equationu ⁿ=P(z) withn ε ℕ,n ≥ 2, andz ε ℂ, whereP(z) is an entire function with infinitely many simple zeros. OnR, the Riemann boundary-value problem for an arbitrary piecewise smooth contour Γ is considered. Necessary and sufficient conditions for its solvability are obtained, and its explicit solution is constructed. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 25–35, January, 2000.